Track transition curve: Difference between revisions

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{{Short description|Mathematically-calculated curve in which a straight section changes into a curve}}

[[Image:Easement curve.svg|240px|thumb|The red [[Euler spiral]] is an example of an easement curve between a blue straight line and a circular arc, shown in green.]]

[[File:CornuSprialAnimation.gif|thumb|Animation depicting evolution of a [[Cornu spiral]] with the tangential circle with the same radius of curvature as at its tip, also known as an [[osculating circle]] (click on thumbnail to observe).]]

[[Image:Parabolic transition curve.JPG|thumb|240px|This sign aside a railroad (between [[Ghent]] and [[Bruges]]) indicates the start of the transition curve. A [[parabolic curve]] (''POB'') is used.]]

A '''track transition curve''', or '''spiral easement''', is a mathematically-calculated curve on a section of highway, or [[track (rail transport)|railroad track]], in which a straight section changes into a curve. It is designed to prevent sudden changes in [[centripetal acceleration|lateral (or centripetal) acceleration]]. In plane (viewed from above), the start of the transition of the horizontal curve is at infinite radius, and at the end of the transition, it has the same radius as the curve itself and so forms a very broad spiral. At the same time, in the vertical plane, the outside of the curve is gradually raised until the correct degree of [[cant (road/rail)|bank]] is reached.

[[Image:Parabolic transition curve.JPG|thumb|240px|This sign aside a railroad (between [[Ghent]] and [[Bruges]]) indicates the start of the transition curve. A [[parabolic curve]] (''POB'') is used.]]

If such an easement were not applied, the lateral acceleration of a rail vehicle would change abruptly at one point (the [[tangent|tangent point]] where the straight track meets the curve), with undesirable results. With a road vehicle, the driver naturally applies the steering alteration in a gradual manner, and the curve is designed to permit that by using the same principle.

A '''transition curve''' (also, '''spiral easement''' or, simply, '''spiral''') is a spiral-shaped length of highway or [[track (rail transport)|railroad track]] that is used between sections having different profiles and radii, such as between straightaways ([[tangent]]s) and curves, or between two different curves.<ref name="pwayblog">{{Cite web |last=Constantin |date=2016-07-03 |title=The Clothoid |url=https://pwayblog.com/2016/07/03/the-clothoid/ |access-date=2023-06-07 |website=Pwayblog}}</ref>

{| class="wikitable" style=width:200px

|-

! Centripetal force on vehicles on roads without and with a transition curve

|-

| [[File:Straight-circle.gif|frameless|upright=1.2|center]] Straight sections of road connected directly by a circular arc. |

|-

| [[File:Straight-cornu.gif|frameless|upright=1.2|center]] Straight section connected to a circular arc via a Cornu spiral|

|-

| Comparison of a poorly designed road with no transition curve with the sudden application of centripetal force required to move in a circle versus a well-designed road in which the centripetal acceleration builds up gradually on a Cornu spiral before being constant on the circular arc. The second animation shows the increasing curvature of the transition curve which is able to connect to a circular arc of progressively smaller radius.

|}

In the horizontal plane, the radius of a transition curve varies continually over its length between the disparate radii of the sections that it joins—for example, from infinite radius at a tangent to the nominal radius of a smooth curve. The resulting spiral provides a gradual, eased transition, preventing undesirable sudden, abrupt changes in [[centripetal acceleration|lateral (centripetal) acceleration]] that would otherwise occur without a transition curve. Similarly, on highways, transition curves allow drivers to change steering gradually when entering or exiting curves.

Transition curves also serve as a transition in the vertical plane, whereby the elevation of the inside or outside of the curve is lowered or raised to reach the nominal amount of [[cant (road/rail)|bank]] for the curve.

==History==

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| last = Rankine

| first = William

| authorlinkauthor-link = William John Macquorn Rankine

| title = A Manual of Civil Engineering

| year = 1883

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In the UK, only from 1845, when legislation and land costs began to constrain the laying out of rail routes and tighter curves were necessary, were the principles beginning to be applied in practice.

[[File:Brusio Viaduct (158241421).jpeg|thumb|[[Brusio spiral viaduct]] and railway (Switzerland, built 1908), from above]]

The 'true spiral', whose curvature is exactly linear in arclength, requires more sophisticated mathematics (in particular, the ability to integrate its [[intrinsic equation]]) to compute than the proposals that were cited by Rankine. Several late-19th century civil engineers seem to have derived the equation for this curve independently (all unaware of the original characterization of the curve by [[Leonhard Euler]] in 1744). Charles Crandall<ref>{{cite book

| last = Crandall

Line 47 ⟶ 62:

| year = 1900

| title = Transition Curves for Railways

| booktitlebook-title = Minutes of Proceedings of the Institution of Civil Engineers

| pages = 161–179

| url = https://books.google.com/books?id=aQsAAAAAMAAJ&pg=PA161

}}</ref>

The equivalence of the railroad transition spiral and the [[clothoid]] seems to have been first published in 1922 by Arthur Lovat Higgins.<ref name="higgins"/> Since then, "clothoid" is the most common name given the curve, but the correct name (following standards of academic attribution) is 'the [[Euler spiral]]'.<ref>{{Cite journal |last=Archibald |first=Raymond Clare |author-link=Raymond Clare Archibald |date=June 1917 |title=Euler Integrals and Euler's Spiral--Sometimes called Fresnel Integrals and the Clothoide or Cornu's Spiral |url=http://www.glassblower.info/Euler-Spiral/AMM/AMM-1918.HTML |journal=American Mathematical Monthly |volume=25 |issue=6 |pages=276–282 |via=Glassblower.Info}}</ref>

==Geometry==

{{unreferenced section|date=January 2010}}

While railroad [[track geometry]] is intrinsically [[three-dimensional space|three-dimensional]], for practical purposes the vertical and horizontal components of track geometry are usually treated separately.<ref>{{Cite web |url=http://www.engr.uky.edu/~jrose/RailwayIntro/Modules/Module%206%20Railway%20Alignment%20Design%20and%20Geometry%20REES%202010.pdf |title=Railway Alignment Design and Geometry |last=Lautala |first=Pasi |last2=Dick |first2=Tyler}}</ref><ref>{{Cite book |title=Practical Guide to Railway Engineering |last=Lindamood |first=Brian |last2=Strong |first2=James C. |last3=McLeod |first3=James |publisher=[[American Railway Engineering and Maintenance-of-Way Association]] |year=2003 |chapter=Railway Track Design |chapter-url=http://www.engsoc.org/~josh/AREMA/chapter6%20-%20Railway%20Track%20Design.pdf |archive-url=https://web.archive.org/web/20161130162616/http://www.engsoc.org:80/~josh/AREMA/chapter6%20-%20Railway%20Track%20Design.pdf |archive-date=November 2030, 2016}}</ref>

The overall design pattern for the vertical geometry is typically a sequence of constant grade segments connected by vertical transition curves in which the local grade varies linearly with distance and in which the elevation therefore varies [[quadratic equation|quadratically]] with distance. Here grade refers to the tangent of the angle of rise of the track. The design pattern for horizontal geometry is typically a sequence of straight line (i.e., a [[tangent]]) and curve (i.e. a [[circular arc]]) segments connected by transition curves.

The degree of banking in railroad track is typically expressed as the difference in elevation of the two rails, commonly quantified and referred to as the [[Cant (road/rail)|superelevation]]. Such difference in the elevation of the rails is intended to compensate for the [[centripetal acceleration]] needed for an object to move along a curved path, so that the lateral acceleration experienced by passengers/the cargo load will be minimized, which enhances passenger comfort/reduces the chance of load shifting (movement of cargo during transit, causing accidents and damage).

It is important to note that superelevation is not the same as the roll angle of the rail which is used to describe the "tilting" of the individual rails instead of the banking of the entire track structure as reflected by the elevation difference at the "top of rail". Regardless of the horizontal alignment and the superelevation of the track, the individual rails are almost always designed to "roll"/"cant" towards gage side (the side where the wheel is in contact with the rail) to compensate for the horizontal forces exerted by wheels under normal rail traffic.

Line 85 ⟶ 100:

| last =Simmons

| first =Jack

| authorlink =

|author2=Biddle, Gordon

| title =The Oxford Companion to British Railway History

| publisher =Oxford University Press

| year =1997

| ___location =

| pages =

| url =

| doi =

| isbn = 0-19-211697-5}}

*{{cite book

| last =Biddle

| first =Gordon

| authorlink =

| title =The Railway Surveyors

| publisher =Ian Allan

| year =1990

| ___location = Chertsey, UK

| pages =

| url =

| doi =

| isbn = 0-7110-1954-1}}

*{{cite book

| last =Hickerson

| first =Thomas Felix

| authorlink =

| title =Route Location and Design

| publisher =McGraw Hill

| year =1967

| ___location =New York

| pages =

| url =

| doi =

| isbn = 0-07-028680-9}}

*{{cite book

| last =Cole

| first =George M

| authorlink =

|author2=and Harbin |author3=Andrew L

| title =Surveyor Reference Manual

Line 128 ⟶ 129:

| year =2006

| ___location =Belmont, CA

| url =

| doi =

| isbn = 1-59126-044-2

| page =16}}